Question

Let f(x) = x², g(x) = 2ˣ, then solve the equation (fog) (x) = (gof) (x).

Answer 1

Answer:    x = 0 & x = 2

Given that

f(x) = x², g(x) = 2ˣ

We have to solve

(fog) (x) = (gof) (x)

⇒

f [g(x)] = g [f(x)]

f [ 2ˣ ] = g [ x² ]

⇒

$(2^{x})^{2}=2^x^{2}\\\\2^{2x}=2^x^{2}\\\\2x=x^{2}\\\\x^{2}=2x\\\\x^{2}-2x=0\\\\x(x-2)=0\\\\x=0ANDx=2$

x = 0 & x = 2

Answer 2

Answer:

x = 0 or x = 2

Step-by-step explanation:

It is given that

f(x) = x² , g(x) = $2^{x}$

i ) ( fog )(x) = f[g(x)]

= f($2^{x}$)

= ( $2^{x}$)²

= $2^{2x}$ ---( 1 )

ii ) (gof)(x)

= g[f(x)]

= g(x²)

= $2^{x^{2} }$-----( 2 )

( 1 ) = ( 2 ) given ,

($2^{[x^{2} }$) = ($2^{x}$)²

⇒ x² = 2x

⇒ x² - 2x = 0

⇒ x( x - 2 ) = 0

x = 0 or x -2 = 0

x =0 or x =2

......